Suppose that a function is described by a formula. If it has an inverse that is a function, we proceed as follows to find a formula for f−1.
Example6
Determine whether the function f(x)=2x−3 is one-to-one, and if it is, find a formula for f−1(x).
Solution
The graph of f is shown at left. It passes the horizontal-line test. Thus it is one-to-one and its inverse is a function. We also proved that f is one-to-one in Example3. We find a formula for f−1(x).
The output is 7. Now we use 7 for the input in the inverse:
f−1(7)=7+32=102=5.
The function f takes the number 5 to 7. The inverse function f−1 takes the number 7 back to 5.
Example7
Graph
f(x)=2x−3andf−1(x)=x+32
using the same set of axes. Then compare the two graphs.
Solution
The graphs of f and f−1 are shown at left. The solutions of the inverse function can be found from those of the original function by interchanging the first and second coordinates of each ordered pair.
When we interchange x and y in finding a formula for the inverse of f(x)=2x−3, we are in effect reflecting the graph of that function across the line y=x. For example, when the coordinates of the y-intercept, (0, −3), of the graph of f are reversed, we get the x-intercept, (−3, 0), of the graph of f−1. If we were to graph f(x)=2x−3 in wet ink and fold along the line y=x, the graph of f−1(x)=(x+3)/2 would be formed by the ink transferred from f.
Example8
Consider g(x)=x3+2.
Determine whether the function is one-to-one.
If it is one-to-one, find a formula for its inverse.
Graph the function and its inverse.
Solution
The graph of g(x)=x3+2 is shown at left. It passes the horizontal-line test and thus has an inverse that is a function. We also know that g(x) is one-to-one because it is an increasing function over its entire domain.